Bond-Length and Area Strain in Binary Alloy Surfaces

Transition-metal alloys such as AuPt, AgPt, AuPd, and AuCu [1,3] exhibit local geometric distortions arising from lattice mismatch and alloy-induced atomic rearrangements, resulting in heterogeneous atomic-scale strain fields [1,2]. We present a framework for quantifying local strain in binary fcc(111) surface alloys. The approach builds on the strain-maps methodology [3] and, although formulated here for binary metallic alloys, can in principle be extended to other material classes through appropriate modification of the coordination and reference-structure definitions. To characterize the net local strain arising from both short- and long-range structural effects, the strain maps combine different geometric descriptors: (i) nearest-neighbor bond-length strain, which probes pairwise distortions, and (ii) atom-resolved Voronoi area strain, which quantifies variations in the local atomic surface environment.

Atomic structures, periodic boundary conditions, database access, and minimum-image geometric operations were handled using the Atomic Simulation Environment (ASE) [4].

Because alloy surfaces may exhibit small out-of-plane corrugations, a representative surface normal vector n was estimated from the surface atomic coordinates by singular value decomposition (SVD) after median-height filtering. This procedure reduces the influence of relaxation-induced height fluctuations while preserving the average surface orientation.

Let v_ij denote the minimum-image interatomic vector between a surface atom i and a candidate neighboring atom j. Its projection onto the best-fit surface plane is

v_ij^|| = v_ij − (v_ij · n)n,

with projected bond distance

d_ij = ||v_ij^|||.

For each surface atom i, candidate neighboring atoms j were ranked according to d_ij. The six closest neighbors were retained for each surface atom, corresponding to the first coordination shell of an ideal fcc(111) surface, where each surface atom has six in-plane nearest neighbors in a hexagonal arrangement. Restricting the analysis to these six neighbors isolates the first coordination shell while excluding contributions from higher coordination shells.

The net local bond strain was defined relative to the pair-dependent equilibrium distance d_ab^0, where a and b denote the chemical identities of the bonded atoms:

ε_ij^bond (%) = 100(d_ij − d_ab^0)/d_ab^0.

For homonuclear pairs, d_ab^0 corresponds to the elemental equilibrium bond length, whereas for heteronuclear pairs it is defined as the average of the corresponding elemental equilibrium bond lengths.

To characterize the local atomic area associated with each surface site, the slab was periodically replicated in the in-plane directions, and the projected atomic coordinates were represented in a two-dimensional basis defined by the surface lattice vectors. A Voronoi tessellation was then constructed from the periodically repeated projected positions, retaining only cells associated with atoms belonging to the central simulation tile.

For each surface atom i, the projected Voronoi area was computed from the ordered vertices of the corresponding polygon using the shoelace formula,

A_i^2D = (1/2)|Σ_k(x_k y_k+1 − y_k x_k+1)|,

where (x_k,y_k) are the ordered polygon vertices and m is the number of polygon vertices (equivalently, polygon edges). For nearly ideal fcc(111) surfaces, Voronoi cells are predominantly hexagonal, such that m ≈ 6, although local chemical disorder and strain fluctuations may produce polygons with different numbers of vertices.

Because the tessellation is constructed from projected coordinates, the corresponding physical surface area was corrected according to the best-fit surface orientation,

A_i = A_i^2D / |n_z|,

where n_z is the out-of-plane component of n. The local area strain was then defined as

ε_i^area (%) = 100(A_i − A_i^0)/A_i^0,

where A_i^0 is the reference Voronoi area associated with the chemical identity of atom i, obtained from independently relaxed pure-metal slabs using the same Voronoi-based methodology and computational settings.

The resulting framework provides both pair-resolved bond distortions and atom-resolved area distortions, enabling quantitative characterization of the net local structural heterogeneity induced by alloying and atomic-size mismatch in binary fcc(111) metallic alloy surfaces [3].

References

[1] M. S. Ozório, M. F. Nygaard, and J. Rossmeisl. “Competitive strain modulation of oxygen reduction reaction in monolayer binary alloy surfaces”. Journal of Catalysis 2025, 115988.

[2] M. S. Ozório, M. F. Nygaard, A. S. Petersen, N. J. Belm, and J. Rossmeisl. “Self-induced long-range surface strain improves oxygen reduction reaction”. Journal of Catalysis 2024, 115484.

[3] M. S. Ozório and J. Rossmeisl. “Stable and Active AuCu and AuPd Electrocatalysts for Platinum-Free Oxygen Reduction Reaction: Design Principles from Ligand-Strain Coupling and Gold Surface Segregation”. ChemRxiv preprint 2026.

[4] A. H. Larsen et al. “The Atomic Simulation Environment – A Python library for working with atoms”. Journal of Physics: Condensed Matter 27 (2017), 273002.